| L(s) = 1 | − 6.94·2-s − 79.7·4-s − 125·5-s − 343·7-s + 1.44e3·8-s + 868.·10-s − 3.45e3·11-s + 2.79e3·13-s + 2.38e3·14-s + 179.·16-s − 2.00e4·17-s + 1.80e4·19-s + 9.96e3·20-s + 2.39e4·22-s − 3.50e4·23-s + 1.56e4·25-s − 1.94e4·26-s + 2.73e4·28-s + 6.36e4·29-s + 1.29e5·31-s − 1.85e5·32-s + 1.39e5·34-s + 4.28e4·35-s + 5.71e5·37-s − 1.25e5·38-s − 1.80e5·40-s + 4.92e4·41-s + ⋯ |
| L(s) = 1 | − 0.614·2-s − 0.622·4-s − 0.447·5-s − 0.377·7-s + 0.996·8-s + 0.274·10-s − 0.782·11-s + 0.352·13-s + 0.232·14-s + 0.0109·16-s − 0.989·17-s + 0.603·19-s + 0.278·20-s + 0.480·22-s − 0.600·23-s + 0.199·25-s − 0.216·26-s + 0.235·28-s + 0.484·29-s + 0.780·31-s − 1.00·32-s + 0.607·34-s + 0.169·35-s + 1.85·37-s − 0.370·38-s − 0.445·40-s + 0.111·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 125T \) |
| 7 | \( 1 + 343T \) |
| good | 2 | \( 1 + 6.94T + 128T^{2} \) |
| 11 | \( 1 + 3.45e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 2.79e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.00e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.80e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 3.50e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 6.36e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.29e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 5.71e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 4.92e4T + 1.94e11T^{2} \) |
| 43 | \( 1 + 6.66e4T + 2.71e11T^{2} \) |
| 47 | \( 1 - 9.54e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.48e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.07e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 7.19e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.83e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 3.15e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.07e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.93e5T + 1.92e13T^{2} \) |
| 83 | \( 1 - 2.97e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.06e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.20e6T + 8.07e13T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.915238242619748705969498290433, −9.011314951147461550261772525465, −8.168124424593295538463533754179, −7.40366469294496814789647225578, −6.11969873502788046354076066361, −4.84706491958949532479442136088, −3.94827312145064512393456036823, −2.57843482301217703064487931497, −0.981533502755264344051046385424, 0,
0.981533502755264344051046385424, 2.57843482301217703064487931497, 3.94827312145064512393456036823, 4.84706491958949532479442136088, 6.11969873502788046354076066361, 7.40366469294496814789647225578, 8.168124424593295538463533754179, 9.011314951147461550261772525465, 9.915238242619748705969498290433
